The first words of this thread: Fractals are more than pretty little pictures.

It is imo an entirely new geometry that reveals the natural world.
Other geometries are piecemeal in comparison;
Euclidean geometry. ...In several ancient cultures there developed a form of geometry suited to the relationships between lengths, areas, and volumes of physical objects. it did discover the golden mean and led to avogadro's number HINTING AT FRACTAL GEOMETRY
Analytic geometry. ...
Projective geometry. ...
Differential geometry. ...
Non-Euclidean geometries. ...
Topology.

BUT FRACTAL IS MORE THAN NON EUCLIDIAN GEOMETRY. It was discovered in 1976 and has not been expanded to its full potential imo.
I foresee a relationship to quantum field wave variations.
Arthur C Clark would repeat "fractals are the fingerprint of God".